For an extension of the results on nonlinear equations with initial data in adapted spaces, one would have to find a suitable definition on compact manifolds. On the other hand, to prove global results it seems necessary to work with spaces of initial data which are invariant under more general FIOs. Indeed, the solution operator to the linear part of (REF ) is a Fourier integral operator, an observation which goes back to Lax [1]}. This motivated the pioneering works by Seeger–Sogge–Stein [2]} and Mockenhaupt–Seeger–Sogge [3]} on the fixed-time and space-time mapping properties of FIOs. It is unclear whether \(\mathcal {B}^s_{p,2,2}(\mathbb {R}^{d})\)  is invariant under more general FIOs, but \(\mathcal {H}^{s,p}_{FIO}(\mathbb {R}^{d})\)  is. Moreover, one could solve nonlinear wave equations with initial data in \(\mathcal {H}^{s,p}_{FIO}(\mathbb {R}^{d})\)  in the same manner as we do for \(\mathcal {B}^s_{p,2,2}(\mathbb {R}^{d})\) .
